Hello Ross, I'm not an Oak framer or joiner so this is outside my area of expertise, but a lot will depend on what you mean by "well seasoned".
The oak framing boys that I know always work in wet Oak, and by and large they source their big wet Oak sections from France in order to get the quality they're looking for.
Four inch thick Oak is close to the limit for kilning, if that's what you mean by well seasoned, and the price is always high to reflect the very extended processing time with thicker sections. Furthermore it's been my experience that quality problems with kilned Oak also go up in step with material thickness. Logical really as the temptation for the timber mill to accelerate the processing time rises with thickness, and that results in checking, honeycombing, and the dreaded "yellow patches", which unfortunately seems to be on the increase with Oak everywhere.
Given your location I'd visit EWT in person, talk it through with them, and physically inspect their stock. EWT carry both air dried and kilned Oak in 100mm thickness. If you do go for kilned then I'd suggest you ask if they kilned the timber or if they bought it in ready kilned. They have had some problems with yellow patches recently but never on the timber they kiln themselves.
Good luck!